When \(Z\)  is negative, classically one expects a pion condensate to form,
but the solution at large \(N\)  shows that instead a quantum pion liquid (Q\(\pi \) L) forms .
While this is rigorous at large \(N\) , as it arises from the double pole at \(k_c \ne 0\)  for
the would be Goldstone modes, it is very likely that there is a Q\(\pi \) L for all \(N > 2\) 
[1]}. I also assume that the quantum fluctuations are sufficiently strong
so that a Q\(\pi \) L forms for massive pions.
